As he kicked off his post-election thank-you tour in Cincinnati, President-elect Donald Trump announced — sort of — that he was naming U.S. Marine Corps Gen. James N. Mattis as defense secretary. But he told the crowd and the TV cameras not to let it out of the room.

“We are going to appoint ‘Mad Dog’ Mattis as our secretary of defense, but we are not announcing it until Monday, so don’t tell anybody,” Mr. Trump said, referring to the general by his nickname.

“They say he is the closest thing to General George Patton that we have, and it is about time, it is about time,” he said.

The emergence of Gen. Mattis could create a “complex chemistry” between the Pentagon and the White House, analysts already were saying upon the first media reports of the Mattis choice Thursday afternoon.

For the general to assume the post, he would need Congress to waive a federal law that bars anyone who has been on active duty in the last seven years from becoming defense secretary. Gen. Mattis, 66, retired as commander of the U.S. Central Command in 2013 after a more than 40-year career in the Marine Corps.

Despite needing that waiver, the general, who was unceremoniously forced out of the Pentagon by the Obama administration over his opposition to the Iran nuclear deal, was already finding strong support Thursday.

The emergence of retired Marine Corps Gen. Mattis as the leading candidate to head President-elect Donald Trump’s Defense Department could create a “complex chemistry” between the Pentagon and the White House, analysts say.

Gen. Mattis, who was unceremoniously forced out of the Pentagon by the Obama administration over his opposition to the Iran nuclear deal, is now said to be Mr. Trump’s pick for defense secretary. The Washington Post and several media outlets reported the choice Thursday, citing unnamed sources inside the Trump transition team.

Mr. Trump’s transition team did not confirm the reports, and said Mr. Trump was still considering his options. “No decision has been made yet with regard to Secretary of Defense,” transition team spokesman Jason Miller said on Twitter Thursday afternoon.

Officials from both the House and Senate Armed Services committees declined to comment on the report, noting the Trump team has yet to confirm the former four-star general as the definite pick to lead the Pentagon. Although he will need a waiver because of his recent service in the military, the general is already finding strong support.

“The public will quickly recognize a person of deep character, strong intellect and administrative skills that go beyond the courage and leadership he has shown on the battlefield,” said Paul McHale, a former assistant defense secretary in the George W. Bush administration, and before that a three-term Democratic congressman.

Mr. McHale, who served with Gen. Mattis as platoon commanders in the Marines, said his military experience — both on and off the battlefield — will pay huge dividends as Defense Department chief.

“Jim knows the building cold,” Mr. McHale said. “Just as he has led quite successfully on the battlefield, his leadership in the Pentagon will be extraordinary.”

If confirmed, Gen. Mattis would be the first ex-military officer to head the Pentagon since President Harry Truman selected George Marshall for the post in 1950.

The blunt-spoken Gen. Mattis has long been considered Mr. Trump’s top candidate for the Pentagon job after the president-elect’s initial pick, retired Army Gen. Jack Keane, declined the nomination. The contenders list has also included former CIA Director and retired Army Gen. David Petraeus and Stephen Hadley, who served as national security adviser for Mr. Bush.

Gen. Mattis, nicknamed “Mad Dog,” could face a challenge in working with another general at the top of Mr. Trump’s national security team — retired Army Lt. Gen. Michael Flynn, who will be Mr. Trump’s national security adviser.

Gen. Flynn, who campaigned heavily for Mr. Trump, has gained a reputation for “being somewhat provocative and ‘disruptive’” during his time in uniform, Michael O’Hanlon, a senior national security analyst with the Brookings Institution in Washington, said.

Given Gen. Flynn’s reputation and Gen. Mattis’ penchant for bluntly speaking his mind on national security issues, “there could be some complex chemistry between him and Mattis,” Mr. O’Hanlon said. “I’m not sure that’s bad, but it’s possible.”

If nominated, Gen. Mattis would require a congressional waiver from federal law that bars retired military officials for seven years from taking civilian government posts. He retired from the Marine Corps in May 2013.

But given Gen. Mattis’ strong reputation on Capitol Hill, along with the fact that the GOP holds majorities in both the House and Senate, securing a waiver will not be difficult, Mr. O’Hanlon said. Fox News reported Thursday that lawmakers on Capitol Hill are already working up draft legislation specifically allowing Gen. Mattis to take the post.

Rep. Mac Thornberry, Texas Republican and chairman of the House Armed Services Committee, called Gen. Mattis “an excellent selection” and signaled that he’d help push the needed waiver.

“I will work with my colleagues in the coming days to clear the way for his confirmation by the Senate,” he said.

Sen. Dan Sullivan, Alaska Republican and a member of that chamber’s Armed Services Committee, called Gen. Mattis an “inspired choice” and, as a former Marine, ended his statement with “Semper Fi!”

He added that he also would “look forward to working with my Senate colleagues on both sides of the aisle in ensuring the swift confirmation of General Mattis, including any statutory waivers required by the Congress.”

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